$AdS_2$ holography is (non-)trivial for (non-)constant dilaton

TL;DR

This work shows that AdS$_2$ holography in generic two-dimensional dilaton gravity with a Maxwell field is trivial for constant dilaton boundary conditions, as canonical charges vanish and the quantum gravity partition function equals unity. The authors establish the robustness of this result against several generalizations, including looser boundary conditions and non-linear couplings, by both canonical and path-integral analyses. They then isolate a non-trivial AdS$_2$ sector in the charged Jackiw–Teitelboim model with linear dilaton boundary conditions, where a Virasoro algebra with central charge $c=24k\bar X/(2\pi)$ emerges and entropy computed via Euclidean methods, Wald, and Cardy formulas matches in the appropriate limits. These findings imply that genuine AdS$_2$ holography in a pure two-dimensional setting requires a non-constant dilaton, and point toward rich structure when linear dilaton sectors are considered. The results have implications for understanding microscopic entropy and the role of boundary conditions in holography for $AdS_2$ spacetimes.

Abstract

We study generic two-dimensional dilaton gravity with a Maxwell field and prove its triviality for constant dilaton boundary conditions, despite of the appearance of a Virasoro algebra with non-zero central charge. We do this by calculating the canonical boundary charges, which turn out to be trivial, and by calculating the quantum gravity partition function, which turns out to be unity. We show that none of the following modifications changes our conclusions: looser boundary conditions, non-linear interactions of the Maxwell field with the dilaton, inclusion of higher spin fields, inclusion of generic gauge fields. Finally, we consider specifically the charged Jackiw--Teitelboim model, whose holographic study was pioneered by Hartman and Strominger, and show that it is non-trivial for certain linear dilaton boundary conditions. We calculate the entropy from the Euclidean path integral, using Wald's method and exploiting the chiral Cardy formula. The macroscopic and microscopic results for entropy agree with each other.